Analysis of generalized QBD queues with matrix-geometrically distributed batch arrivals and services

In a quasi-birth–death (QBD) queue, the level forward and level backward transitions of a QBD-type Markov chain are interpreted as customer arrivals and services. In the generalized QBD queue considered in this paper, arrivals and services can occur in matrix-geometrically distributed batches. This paper presents the queue length and sojourn time analysis of generalized QBD queues. It is shown that, if the number of phases is N, the number of customers in the system is order-N matrix-geometrically distributed, and the sojourn time is order-(Formula presented.) matrix-exponentially distributed, just like in the case of classical QBD queues without batches. Furthermore, phase-type representations are provided for both distributions. In the special case of the arrival and service processes being independent, further simplifications make it possible to obtain a more compact, order-N representation for the sojourn time distribution. © 2015 Springer Science+Business Media New Yor

Call-Routing Schemes for Call-Center Outsourcing

Companies may choose to outsource parts, but not all, of their call-center operations. In the course of studying contact centers in the telecommunications and financial services industries, we have observed the following (apparently) common scheme. A company classifies its customers as high- or low-value, serving the former with their “in house” operations and routing the latter to an outsourcer. Typically, the company imposes service-level constraints on the time each type of customer waits on hold. This paper considers four schemes for routing low-value calls between the client company and the outsourcer. These schemes vary in the complexity of their routing algorithms, as well as the sophistication of the telephone and information technology infrastructure they require of the two operations. For three of these schemes, we provide a direct characterization of system performance. For the fourth, most complex, scheme we provide performance bounds for the important special case in which the service requirements of high- and low-value callers are the same. These results allow us to systematically compare the performance of the various routing schemes. Our results suggest that, for clients with large outsourcing requirements, the simpler schemes that require little client-outsourcer coordination can perform very well